Quantum Probability: Past, Present and Future

Quantum walks are considered to be quantum counterparts of classical random walks. With the advancements in quantum information theory, different version of quantum walks and quantum random walks have been developed for application in quantum algorithms and mimicking/simulating dynamics of various physical systems. In this talk I will introduce both, the discrete-time and continuous-time versions of quantum walks and show the ways of controlling their probability distribution. I will also present ways of introducing randomness in the dynamics to see the transition from quantum walks to quantum random walks which can be used to simulating "localization" and other interesting phenomenon observed in the physical systems due to quantum interference. Speaker: Franco Fagnola, Poltecnico di Milano, Italy Title: On the decoherence-free subalgebra and the set of fixed points Abstract: We discuss the relationships between the decoherence-free subalgebra and the structure of the fixed point subalgebra of a quantum Markov semigroup on B(h) with a faithful normal invariant state. We show that atomicity of the decoherence-free subalgebra is equivalent to typical splittings of B(h) into the a subalgebra where maps of the semigroup acts as endomorphisms and a remainder space vanishing for long times. Moreover, we characterize the set of reversible states. References: [1] J. Deschamps, F. Fagnola, E. Sasso and V. Umanita. Structure of Uniformly Continuous Quantum Markov Semigroups. Rev. Math. Phys. 28 (2016), 1650003-1 -1650003-32. [2] F. Fagnola, E. Sasso and V. Umanita. Relationships between the decoherencefree algebra and the set of fixed points. In preparation. We discuss the relationships between the decoherence-free subalgebra and the structure of the fixed point subalgebra of a quantum Markov semigroup on B(h) with a faithful normal invariant state. We show that atomicity of the decoherence-free subalgebra is equivalent to typical splittings of B(h) into the a subalgebra where maps of the semigroup acts as endomorphisms and a remainder space vanishing for long times. Moreover, we characterize the set of reversible states. References: [1] J. Deschamps, F. Fagnola, E. Sasso and V. Umanita. Structure of Uniformly Continuous Quantum Markov Semigroups. Rev. Math. Phys. 28 (2016), 1650003-1 -1650003-32. [2] F. Fagnola, E. Sasso and V. Umanita. Relationships between the decoherencefree algebra and the set of fixed points. In preparation. Speaker: Paolo Gibilisco, Centro Vito Volterra, Rome (Italy) Title: Integrability of the generalized Proudman-Johnson equation, alpha-connections and the geodesics of the L spheres. Abstract: Khesin, Lenells, Misiolek and Preston have shown how to define the alphaconnections on Dens(M), the space of smooth densities on a compact manifold. They do this looking at Dens(M) as a suitable quotient of the diffeomorphism group. In this setting they were able to prove that the equations of alpha-geodesics coincide with the generalized Proudman–Johnson equations. They also discuss, in some cases, integrability and complete integrability of the differential equations. The purpose of this presentation is to expose their results and to discuss a possible link with the approach to alpha-connections via the Amari-Chentsov embedding in the L spheres, due to Gibilisco and Isola. I will close my talk formulating some conjectures which possibly the L approach can help to prove (in collaboration with Nihat Ay). Speaker: Jaeseong Heo, Hanyang University, Seoul, South Korea Title: Hypercyclicity in operator algebras and operator theory Abstract: We discuss a notion of q-frequent hypercyclicity of linear maps and derive a sufficient condition for a linear map to be q-frequently hypercyclic in the strong operator topology. We study q-frequent hypercyclicity of tensor products and direct sums of operators. We discuss the hypercyclicity and supercyclicity for operator matrices in the class S consisting 2 x 2 operator matrices with (1, 2)-entries having closed range. Under some conditions, we find the necessary and sufficient conditions for 2 x 2 operator matrices in some class S for which Weyl's theorem, Browder's theorem, a-Weyl's theorem or aBrowder's theorem hold. We discuss a notion of q-frequent hypercyclicity of linear maps and derive a sufficient condition for a linear map to be q-frequently hypercyclic in the strong operator topology. We study q-frequent hypercyclicity of tensor products and direct sums of operators. We discuss the hypercyclicity and supercyclicity for operator matrices in the class S consisting 2 x 2 operator matrices with (1, 2)-entries having closed range. Under some conditions, we find the necessary and sufficient conditions for 2 x 2 operator matrices in some class S for which Weyl's theorem, Browder's theorem, a-Weyl's theorem or aBrowder's theorem hold. Speaker: Un Cig Ji, Chungbuk University, South Korea Title: Wick Calculus for Admissible White Noise Operators and Applications Abstract: We first discuss the space of all admissible white noise operators as a commutative *-algebra with respect to the Wick product and then, secondly, we study several implementation problems in terms of quantum white noise derivatives. Finally, solutions of implementation problems are applied to a quantum extension of Girsanov theorem for quantum stochastic processes. This talk is based on a series of joint works with Nobuaki Obata. Speaker: Martin Lindsay, Lancaster University, Lancaster, UK Title: The φ-conditional expectation, Markovian cocyces and the 4-semigroups approach Abstract: In this talk I shall highlight some of Luigi Accardi's contributions to quantum probability.